If the vectors and are such that $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{b}}$ form a right handed system, then $\overrightarrow{\mathrm{c}}$ is

• $\mathrm{z}\hat{\mathrm{i}} - \mathrm{x}\hat{\mathrm{k}}$

• $\overrightarrow{0}$

• $\mathrm{y}\hat{\mathrm{j}}$

• $- \mathrm{z}\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{k}}$

The equation of straight line passing through the point (a, b, c) and parallel to Z-axis, is

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{0}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{0}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{1}$

23.

If the equation of the locus of a point equidistant from the points (a₁, b₁) and (a₂, b₂) is (a₁ - a₂)r + (b₁ - b₂)y + c = 0, then the value of 'c' is

• $\frac{1}{2}\left({{\mathrm{a}}_2}^2 + {{\mathrm{b}}_2}^2 - {{\mathrm{a}}_1}^2 - {{\mathrm{b}}_1}^2\right)$

• ${{\mathrm{a}}_1}^2 - {{\mathrm{a}}_2}^2 + {{\mathrm{b}}_1}^2 - {{\mathrm{b}}_2}^2$

• $\frac{1}{2}\left({{\mathrm{a}}_1}^2 + {{\mathrm{a}}_2}^2 + {{\mathrm{b}}_1}^2 + {{\mathrm{b}}_2}^2\right)$

• $\sqrt{{{\mathrm{a}}_1}^2 + {{\mathrm{b}}_1}^2 - {{\mathrm{a}}_2}^2 - {{\mathrm{b}}_2}^2}$

A tetrahedron has vertices at 0(0, 0, 0), A(1, 2, 1), B(2, 1, 3) and C(- 1, 1, 2). Then, the angle between the faces OAB and ABC will be

• ${\cos }^{-1}\left(\frac{19}{35}\right)$

• ${\cos }^{-1}\left(\frac{17}{31}\right)$

• $30°$

• $90°$

Distance between parallel planes 2x - 2y + z + 3 = 0 and 4x - 4y + 2z + 5 = 0, is

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{6}$

The vector $\hat{\mathrm{i}} + \mathrm{x}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is rotated through an angle $\mathrm{\theta }$ and doubled in magnitude, then it becomes $4\hat{\mathrm{i}} + \left(4\mathrm{x} - 2\right)\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$. Then, the values of x are

• $- \frac{2}{3}, 2$

• $\frac{1}{3}, 2$

• $\frac{2}{3}, 0$

• 2, 7

If a = (1, - 1) and b = (- 2, m) are two collinear vectors, then m is equal to

• 4

• 3

• 2

• 0

Given two vectors $\hat{\mathrm{i}} - \hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{i}} + 2\hat{\mathrm{j}}$, the unit vector coplanar with the two vectors and perpendicular to first, is

• $\frac{1}{\sqrt{2}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\frac{1}{\sqrt{5}}\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\pm \frac{1}{\sqrt{2}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• None of these

For any vector a $\hat{\mathrm{i}} \times \left(\mathrm{a} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{j}} \times \left(\mathrm{a} \times \hat{\mathrm{j}}\right) + \hat{\mathrm{k}} \times \left(\mathrm{a} \times \hat{\mathrm{k}}\right)$ is equal to

• 2a

• 3a

• - 2a

• a

If the a, band c form the sides BC, CA and AB respectively, of a $∆$ABC, then

• $\mathrm{a} . \mathrm{b} + \mathrm{b} . \mathrm{c} + \mathrm{c} . \mathrm{a} = 0$

• $\mathrm{a} \times \mathrm{b} = \mathrm{b} \times \mathrm{c} = \mathrm{c} \times \mathrm{a}$

• $\mathrm{a} . \mathrm{b} = \mathrm{b} . \mathrm{c} = \mathrm{c} . \mathrm{a} = 0$

• $\mathrm{a} \times \mathrm{b} + \mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a}$